My opinion would be 'depends'. Which monks and nuns are you talking about specifically? What forms of exercise? How much exercise?
I've seen variations in groups and individuals.
Of monasteries I've visited: some did prostrations as both a religious/physical health exercises, some went for hikes, some didn't do anything really. Some people had regular yoga /tai chi/qi gong practices even. One nun liked to ride a bicycle when it was available. One monk I knew would do some pushups in the morning. Where I lived it was in the mountains so to get anywhere you'd automatically get some exercises just going place to place.
Of the monasteries I saw where monks were playing soccer every evening: they were located in rural parts of India and/or were predominately populated by kids and teenagers. Who were either rabjung or getsul. They had weekends 'off' and would play as well. The attrition rate at monasteries are so high and the reasons the kids are there are varied...but most often it is access to education/food/care until they are old enough to leave. Basic education with some morals instilled. They are also kids and usually need to run out their energy more than an adult.
I don't know about China/Taiwan but from the examples given it seems like they are some institutionalized 'physical fitness' ?
Huseng: I have seen that training rule about not running in a vinaya...but is that about just randomly running place to place or about jogging/running for fitness- with mindfulness and intent? I imagine seeing a monk walking about, doing some business and then bolting off would seem odd, like if they were running to catch a crosswalk or a bus and being flustered...but seeing a monastic on a track exercising wouldn't really be so strange to me. I wonder what the intent behind not running was in the vinaya? Does what you are referencing specify reasons/circumstances/give an example to put it in context? I wonder how popular, if at all, jogging/running for fitness was in the Buddha's time. 'Inappropriate running' seems like a better approach than a vague 'no running'. Even if that were the case there are still many forms of exercise that do not involve running at all!
Porkchop: the "marathon" monks of Hiei don't really "run". It more like a quick mindful walk. They just go a distance near that of a marathon. It is something of a misnomer since it isn't a marathon at all...but mountain/mandala circumambulation.